Question

What does the transformation f(x)↦3f(x) do to the graph of f(x)?
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Answer 1

**Question:** What does the transformation \( f(x) \mapsto 3f(x) \) do to the graph of \( f(x) \)?

**Key Idea:** The transformation \( f(x) \mapsto 3f(x) \) is a vertical scaling (or stretching) of the graph of the function \( f(x) \). Specifically, it multiplies the output values (y-values) of the function by a factor of 3.

**Solution:**
When you transform the function \( f(x) \) into \( 3f(x) \), every point on the graph of \( f(x) \) is affected as follows:
- For every \( x \) in the domain, the new output is three times the original output. This means that if the original function value is \( f(x) \), the transformed function value will be \( 3f(x) \).

Graphs characteristics affected:
- **Vertical Stretch:** The graph is stretched vertically away from the x-axis. Points that were originally at \( (x, f(x)) \) are moved to \( (x, 3f(x)) \).
- **Effect on intercepts:** The x-intercepts of the original graph (where \( f(x) = 0 \)) remain unchanged, since \( 3f(x) = 0 \) is still satisfied by the same \( x \) values.
- **Y-intercept:** The y-intercept at \( (0, f(0)) \) will change to \( (0, 3f(0)) \). If \( f(0) \) was positive, the new y-intercept will be higher; if negative, it will be lower.

In summary, \( f(x) \mapsto 3f(x) \) stretches the graph vertically by a factor of 3, making it taller but not altering its x-intercept locations.